Polynomial Graphs Questions with Solutions

How to use the properties of the polynomial graphs to identify polynomials. Grade 12 math questions with detailed solutions and graphical interpretations are presented.

Question 1

Give four different reasons why the graph below cannot possibly be the graph of the polynomial function \( p(x) = x^4-x^2+1 \). graph of polynomial in question 1.

Solution

The four reasons are:

  1. ) The given polynomial function is even and therefore its graph must be symmetric with respect to the y axis. The given graph is not symmetric with respect to the y axis.
  2. ) The given polynomial function does not have real zeros (discriminant = -3 : negative). The given graph has x intercepts which must corresponds to real zeros.
  3. )The y intercept calculated using \(p(x)\) which is given by: \( p(0) = 0^4 - 0^2 + 1 = 1 \) is positive. The y intercept of the graph is negative.
  4. ) Having a leading coefficient ( = 1) positive and an even degree ( = 4), the polynomial must have a graph with the right and left both rising. In the given graph, they are both falling.

Question 2

Match the polynomial functions to their graphs where all x intercepts are shown. \[ f(x) = (x+1)(x-1)^2(x+2)^2 \] \[ g(x) = -(x+1)(x-1)^4 \] \[ h(x) = (x+1)(x-1)^3(x-3)\] \[ i(x) = (x+1)^2(x-2)^3\] \[ j(x) = (x+1)^2(1-x)(x-2)^2\] \[ k(x) =-(x+1)^2(x-1)^2(x-3)\] graph of polynomial in question 2.

Solution

According to their equations, all 6 given polynomial functions are of degree 5. However their leading coefficients are of different signs. We classify the 6 polynomials into 2 groups: I and II:

Group I - Given polynomials with positive leading coefficients \[ f(x) = (x + 1)(x - 1)^2(x + 2)^2 \] \[ h(x) = (x + 1)(x - 1)^3(x - 3) \] \[ i(x) = (x + 1)^2(x - 2)^3 \]

Having degree 5 (odd) and leading coefficients positive, each of the graphs of the above polynomials \( f, h \) and \( i \) has the following graphical properties:

As \( x \to \infty \), \( y \to \infty \) (the right-hand side of the graph rises)

As \( x \to -\infty \), \( y \to -\infty \) (the left-hand side of the graph falls)

The given graphs in parts a) c) and e) have the above properties with different x intercepts and their multiplicities. Hence

1 - Polynomial \( f(x) = (x + 1)(x - 1)^2(x + 2)^2 \) has a zero of multiplicity 1 at \( x = -1 \), a zero of multiplicity 2 at \( x = 1 \), and a zero of multiplicity 2 at \( x = -2 \), and should correspond to the graph in part e).

2 - Polynomial \( h(x) = (x + 1)(x - 1)^3(x - 3) \) has a zero of multiplicity 1 at \( x = -1 \), a zero of multiplicity 3 at \( x = 1 \), and a zero of multiplicity 1 at \( x = 3 \), and should correspond to the graph in part a).

3 - Polynomial \( i(x) = (x + 1)^2(x - 2)^3 \) has a zero of multiplicity 2 at \( x = -1 \) and a zero of multiplicity 3 at \( x = 2 \), and should correspond to the graph in part c).

Group II - Given polynomials with negative leading coefficients

The polynomial functions \( g \), \( j \), and \( k \), when expanded, have leading coefficients that are negative. \[ g(x) = - (x + 1)(x - 1)^4 \] \[ j(x) = (x + 1)^2(1 - x)(x - 2)^2 \] \[ k(x) = - (x + 1)^2(x - 1)^2(x - 3) \]

Having degree 5 (odd) and negative leading coefficients, each of the graphs of the polynomials \( g \), \( j \), and \( k \) has the following graphical properties: \[ \text{As } x \to \infty, \quad y \to -\infty \quad \text{(the right-hand side of the graph falls)} \] \[ \text{As } x \to -\infty, \quad y \to \infty \quad \text{(the left-hand side of the graph rises)} \]

The given graphs in parts \( b \), \( d \), and \( f \) exhibit the above end-behavior properties but differ in x-intercepts and their multiplicities. Hence:

1 - Polynomial \( g(x) = - (x + 1)(x - 1)^4 \) has a zero of multiplicity 1 at \( x = -1 \), a zero of multiplicity 4 at \( x = 1 \), and should correspond to the graph in part f).

2 - Polynomial \( j(x) = (x + 1)^2(1 - x)(x - 2)^2 \) has a zero of multiplicity 2 at \( x = -1 \), a zero of multiplicity 1 at \( x = 1 \), and a zero of multiplicity 2 at \( x = 2 \), and should correspond to the graph in part d).

3 - Polynomial \( k(x) = - (x + 1)^2(x - 1)^2(x - 3) \) has a zero of multiplicity 2 at \( x = -1 \), a zero of multiplicity 2 at \( x = 1 \), and a zero of multiplicity 1 at \( x = 3 \), and should correspond to the graph in part b).

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